Is 6^17 + 17^6 Divisible by 3 or 7?

[lordy12]

1. 6^17 + 17^6 is divisble by 3 or 7?

Homework Equations

3. 6^1 = 0(mod 3)
(6^1)^17=0(mod 3) so 6^17 is divisible by 3
how do u do this?

 

[matt grime]

17 to any power is prime... No.

What is the stray T in your first line. Where did 61 come from? Why are you raising it to the power 1?

 

[lordy12]

sorry i changed it

 

[matt grime]

Fermat's little theorem.

 

[HallsofIvy]

1. 6^17 + 17^6 is divisble by 3 or 7?

Homework Equations

3. 6^1 = 0(mod 3)
(6^1)^17=0(mod 3) so 6^17 is divisible by 3
how do u do this?

For 3 at least, you don't need anyone as powerful as Fermat! Suppose 6¹⁷+ 17⁶ were divisible by 3. Then we would have 6¹⁷+ 17⁶= 3n for some integer n. Then 17⁶= 3n- 6¹⁷= 3n- 3¹⁷2¹⁷= 3(n- 3¹⁶2¹⁷) which is impossible: since 17 is prime, no power of 17 is divisible by 3.